reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;

theorem Th39:
  A is everywhere_dense iff A` is nowhere_dense
proof
  thus A is everywhere_dense implies A` is nowhere_dense
  proof
    assume A is everywhere_dense;
    then Cl Int A = [#]X;
    then (Cl Int A)` = {}X by Th2;
    then Int (Int A)` = {}X by TDLAT_3:3;
    then Int Cl A` = {} by TDLAT_3:2;
    then Cl A` is boundary;
    hence thesis;
  end;
  assume A` is nowhere_dense;
  then Cl A` is boundary;
  then Int Cl A` = {}X;
  then Int (Int A)` = {}X by TDLAT_3:2;
  then (Cl Int A)` = {}X by TDLAT_3:3;
  then Cl Int A = [#]X by Th2;
  hence thesis;
end;
